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How to Calculate Combinations

Last Updated: November 7, 2023 References

wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. To create this article, volunteer authors worked to edit and improve it over time. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been viewed 121,278 times. Learn more...

Permutations and combinations have uses in math classes and in daily life. Thankfully, they are easy to calculate once you know how. Unlike permutations , where group order matters, in combinations, the order doesn't matter. [1] X Research source Combinations tell you how many ways there are to combine a given number of items in a group. To calculate combinations, you just need to know the number of items you're choosing from, the number of items to choose, and whether or not repetition is allowed (in the most common form of this problem, repetition is not allowed).

Calculating Combinations Without Repetition

Step 1 Consider an example problem where order does not matter and repetition is not allowed.

  • For instance, you may have 10 books, and you'd like to find the number of ways to combine 6 of those books on your shelf. In this case, you don't care about order - you just want to know which groupings of books you could display, assuming you only use any given book once.


  • If you have a calculator available, find the factorial setting and use that to calculate the number of combinations. If you're using Google Calculator, click on the x! button each time after entering the necessary digits.
  • For the example, you can calculate 10! with (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1), which gives you 3,628,800. Find 4! with (4 * 3 * 2 * 1), which gives you 24. Find 6! with (6 * 5 * 4 * 3 * 2 * 1), which gives you 720.
  • Then multiply the two numbers that add to the total of items together. In this example, you should have 24 * 720, so 17,280 will be your denominator.
  • Divide the factorial of the total by the denominator, as described above: 3,628,800/17,280.
  • In the example case, you'd do get 210. This means that there are 210 different ways to combine the books on a shelf, without repetition and where order doesn't matter.

Calculating Combinations with Repetition

  • For instance, imagine that you're going to order 5 items from a menu offering 15 items; the order of your selections doesn't matter, and you don't mind getting multiples of the same item (i.e., repetitions are allowed).


  • This is the least common and least understood type of combination or permutation, and isn't generally taught as often. [9] X Research source Where it is covered, it is often also known as a k -selection, a k -multiset, or a k -combination with repetition. [10] X Research source

{}_{{n+r-1}}C_{{r}}={\frac  {(n+r-1)!}{(n-1)!r!}}

  • If you have to solve by hand, keep in mind that for each factorial , you start with the main number given and then multiply it by the next smallest number, and so on until you get down to 0.
  • For the example problem, your solution should be 11,628. There are 11,628 different ways you could order any 5 items from a selection of 15 items on a menu, where order doesn't matter and repetition is allowed.

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how to solve problem in combination

  • ↑ https://www.calculatorsoup.com/calculators/discretemathematics/combinations.php
  • ↑ https://betterexplained.com/articles/easy-permutations-and-combinations/
  • ↑ https://www.mathsisfun.com/combinatorics/combinations-permutations.html
  • ↑ https://medium.com/i-math/combinations-permutations-fa7ac680f0ac
  • ↑ https://www.quora.com/What-is-Combinations-with-repetition
  • ↑ https://en.wikipedia.org/wiki/Combination
  • ↑ https://www.dummies.com/article/technology/electronics/graphing-calculators/permutations-and-combinations-and-the-ti-84-plus-160925/

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1. Characterize combinations and combinations with repetition.

2. On the plane there are 6 different points (no 3 of them are lying on the same line). How many segments do you get by joining all the points?

3. On a circle there are 9 points selected. How many triangles with edges in these points exist?

4. a) Find out a formula for counting the number of diagonals in a convex n-gon! b) How many diagonals has a 10-gon?

5. In how many ways you can choose 8 of 32 playing cards not considering their order?

The playing cards can be chosen in 10 518 300 ways.

6. A teacher has prepared 20 arithmetics tasks and 30 geometry tasks. For a test he‘d like to use:

7. On a graduation party the graduants pinged their glasses. There were 253 pings. How many graduants came to the party?

8. If the number of elements would raise by 8, number of combinations with k=2 without repetition would raise 11 times. How many elements are there?

9. For which x positive integer stands:

10. Two groups consist of 26 elements and 160 combinations without repetition for k=2 together. How many elements are in the first and how many in the second group?

11. In the confectioners 5 different icecreams are sold. A father would like to buy 15 caps of icecream for his family. In how many ways can he buy the icecream?

12. From how many elements 15 combinations with repetition (k=2) can be made?

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In these lessons, we will learn the concept of combinations, the combination formula and solving problems involving combinations.

Related Pages Permutations Permutations and Combinations Counting Methods Factorial Lessons Probability

What Is Combination In Math?

An arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C. In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.

What Is The Combination Formula?

The number of combinations of n things taken r at a time is written as C( n , r ) .

The following diagram shows the formula for combination. Scroll down the page for more examples and solutions on how to use the combination formula.

Combination Formula

If you are not familiar with the n! (n factorial notation) then have a look the factorial lesson

How To Use The Combination Formula To Solve Word Problems?

Example: In how many ways can a coach choose three swimmers from among five swimmers?

Solution: There are 5 swimmers to be taken 3 at a time. Using the formula:

The coach can choose the swimmers in 10 ways.

Example: Six friends want to play enough games of chess to be sure every one plays everyone else. How many games will they have to play?

Solution: There are 6 players to be taken 2 at a time. Using the formula:

They will need to play 15 games.

Example: In a lottery, each ticket has 5 one-digit numbers 0-9 on it. a) You win if your ticket has the digits in any order. What are your changes of winning? b) You would win only if your ticket has the digits in the required order. What are your chances of winning?

Solution: There are 10 digits to be taken 5 at a time.

The chances of winning are 1 out of 252.

b) Since the order matters, we should use permutation instead of combination. P(10, 5) = 10 x 9 x 8 x 7 x 6 = 30240

The chances of winning are 1 out of 30240.

How To Evaluate Combinations As Well As Solve Counting Problems Using Combinations?

A combination is a grouping or subset of items. For a combination, the order does not matter.

How many committees of 3 can be formed from a group of 4 students? This is a combination and can be written as C(4,3) or 4 C 3 or \(\left( {\begin{array}{*{20}{c}}4\\3\end{array}} \right)\).

  • The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be on the field at any one time?
  • A student need 8 more classes to complete her degree. If she met the prerequisites for all the courses, how many ways can she take 4 classes next semester?
  • There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 man and 2 women. How many different groups can be formed from the office?

How To Solve Word Problems Involving Permutations And Combinations?

  • A museum has 7 paintings by Picasso and wants to arrange 3 of them on the same wall. How many ways are there to do this?
  • How many ways can you arrange the letters in the word LOLLIPOP?
  • A person playing poker is dealt 5 cards. How many different hands could the player have been dealt?

How To Solve Combination Problems That Involve Selecting Groups Based On Conditional Criteria?

Example: A bucket contains the following marbles: 4 red, 3 blue, 4 green, and 3 yellow making 14 total marbles. Each marble is labeled with a number so they can be distinguished.

  • How many sets/groups of 4 marbles are possible?
  • How many sets/groups of 4 are there such that each one is a different color?
  • How many sets of 4 are there in which at least 2 are red?
  • How many sets of 4 are there in which none are red, but at least one is green?

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Combination Word Problems

Exercise 10, exercise 11, exercise 12, solution of exercise 1, solution of exercise 2, solution of exercise 3, solution of exercise 4, solution of exercise 5, solution of exercise 6, solution of exercise 7, solution of exercise 8, solution of exercise 9, solution of exercise 10, solution of exercise 11, solution of exercise 12.


How many different combinations of management can there be to fill the positions of president, vice-president and treasurer of a football club knowing that there are 12 eligible candidates?

How many different ways can the letters in the word "micro" be arranged if it always has to start with a vowel?

How many combinations can the seven colors of the rainbow be arranged into groups of three colors each?

How many different five-digit numbers can be formed with only odd numbered digits? How many of these numbers are greater than 70,000?

How many games will take place in a league consisting of four teams? (Each team plays each other twice, once at each teams respective "home" location)

10 people exchange greetings at a business meeting. How many greetings are exchanged if everyone greets each other once?

How many five-digit numbers can be formed with the digits 1, 2 and 3? How many of those numbers are even?

How many lottery tickets must be purchased to complete all possible combinations of six numbers, each with a possibility of being from 1 to 49?

How many ways can 11 players be positioned on a soccer team considering that the goalie cannot hold another position other than in goal?

How many groups can be made from the word "house" if each group consists of 3 alphabets?

Sarah has 8 colored pencils that are all unique. She wants to pick three colored pencils from her collection and give them to her younger sister. How many different combinations of colored pencils can Sarah make from 8 pencils?

Alice has 6 chocolates. All of the chocolates are of different flavors. She wants to give two of her chocolates to her friend. How many different combinations of chocolates can Alice make from six chocolates?

The order of the elements does matter.

The elements cannot be repeated.

how to solve problem in combination

The words will begin with i or o followed by the remaining 4 letters taken from 4 by 4.

how to solve problem in combination

The order of the elements does not matter.

how to solve problem in combination

n = 5      k = 5

how to solve problem in combination

The odd numbers greater than 70,000 have to begin with 7 or 9. Therefore:

how to solve problem in combination

The elements are repeated.

how to solve problem in combination

If the number is even it can only end in 2.

how to solve problem in combination

Therefore, there are 10 players who can occupy 10 different positions.

how to solve problem in combination

The word house has 5 alphabets. If each new word should have 3 alphabets, then we should use the following formula:

how to solve problem in combination

Substitute the values in this example in the above formula:

how to solve problem in combination

Number of pencils Sarah have = 8

Number of pencils she wants to give to her younger sister = 3

We will use the binomial coefficients formula to determine the number of combinations:

After substitution we will get the number of combinations:

how to solve problem in combination

Hence, Sarah can make 56 combinations of 8 colored pencils given the fact that she can choose 3 at a time.

Number of chocolates Alice have = 6

Number of chocolates she wants to give to her friend= 2

how to solve problem in combination

Hence, Alice can make 15 combinations of 6 chocolates given the fact that she can choose 2.

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Combinatorics, permutations, circular permutations, permutations with repetition, combinations with repetition, pascal’s triangle, combinatorics formulas, permutations word problems, cancel reply.

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I must say everything is superb I wish I could access to most of my maths topics here! Questions are easy to comprehend. Thanks for your help.

in how many ways can be 5 red balls and green balls be arranged in a row of no two green balls can be next to each other?

Great job but explanations are too complex

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Course: precalculus   >   unit 8.

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  1. Solved: Solve each problem using any method. See Examples 1-10....

    how to solve problem in combination

  2. Problem Solving With Combinations

    how to solve problem in combination

  3. Question Video: Solving Problems Involving Permutations and Combinations

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  4. Problem Solved Free Stock Photo

    how to solve problem in combination

  5. Problem Solving With Combinations

    how to solve problem in combination

  6. Founds 12: 2.7 Problem Solving w Permutations & Combinations

    how to solve problem in combination


  1. A Nice Math Problem with Factorial

  2. Can You Solve the P & C Problem in Seconds?


  4. 39. Combination Sum

  5. How to find equivalent resistance of ANY circuit!! #currentelectricity

  6. Finding no. Of ways ||combination problems very important ||permutation and combination chapter||


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